Splittings and automorphisms of relatively hyperbolic groups

TL;DR

This paper investigates the automorphism groups of relatively hyperbolic groups, describing their structure via JSJ decompositions, and characterizes when these groups are infinite, with applications to hyperbolic groups and their splittings.

Contribution

It provides a detailed analysis of Out(G) for relatively hyperbolic groups using JSJ trees, and extends results to automorphisms of parabolic subgroups and conditions for infiniteness.

Findings

Out(G) is described using JSJ decompositions over virtually cyclic or parabolic subgroups.

For toral relatively hyperbolic groups, Out(G) is virtually built from mapping class groups and GL_n(Z) subgroups.

Conditions for Out(G) to be infinite are characterized in terms of splittings and automorphisms of vertex groups.

Abstract

We study automorphisms of a relatively hyperbolic group G. When G is one-ended, we describe Out(G) using a preferred JSJ tree over subgroups that are virtually cyclic or parabolic. In particular, when G is toral relatively hyperbolic, Out(G) is virtually built out of mapping class groups and subgroups of GL_n(Z) fixing certain basis elements. When more general parabolic groups are allowed, these subgroups of GL_n(Z) have to be replaced by McCool groups: automorphisms of parabolic groups acting trivially (i.e. by conjugation) on certain subgroups. Given a malnormal quasiconvex subgroup P of a hyperbolic group G, we view G as hyperbolic relative to P and we apply the previous analysis to describe the group Out(P to G) of automorphisms of P that extend to G: it is virtually a McCool group. If Out(P to G) is infinite, then P is a vertex group in a splitting of G. If P is torsion-free, then Out(P to G) is of type VF, in particular finitely presented. We also determine when Out(G) is infinite, for G relatively hyperbolic. The interesting case is when G is infinitely-ended and has torsion. When G is hyperbolic, we show that Out(G) is infinite if and only if G splits over a maximal virtually cyclic subgroup with infinite center. In general we show that infiniteness of Out(G) comes from the existence of a splitting with infinitely many twists, or having a vertex group that is maximal parabolic with infinitely many automorphisms acting trivially on incident edge groups.